Abstract
The behavior of matter near a quantum critical point is one of the most exciting and challenging areas of physics research. Emergent phenomena such as high-temperature superconductivity are linked to the proximity to a quantum critical point. Although significant progress has been made in understanding quantum critical behavior in some low dimensional magnetic insulators, the situation in metallic systems is much less clear. Here, we demonstrate that NiCoCrx single crystal alloys are remarkable model systems for investigating quantum critical point physics in a metallic environment. For NiCoCrx alloys with x ≈ 0.8, the critical exponents associated with a ferromagnetic quantum critical point are experimentally determined from low temperature magnetization and heat capacity measurements. All of the five exponents (γ T ≈ 1/2, β T ≈ 1, δ ≈ 3/2, νzm ≈ 2, \({\bar {\rm \alpha}}\) T ≈ 0) are in remarkable agreement with predictions of Belitz–Kirkpatrick–Vojta theory in the asymptotic limit of high disorder. Using these critical exponents, excellent scaling of the magnetization data is demonstrated with no adjustable parameters. We also find a divergence of the magnetic Gruneisen parameter, consistent with a ferromagnetic quantum critical point. This work therefore demonstrates that entropy stabilized concentrated solid solutions represent a unique platform to study quantum critical behavior in a highly tunable class of materials.
Similar content being viewed by others
Introduction
The connection between quantum fluctuations in the vicinity of a quantum critical point (QCP) and emergent ground states, such as high-temperature superconductivity, remains a topic of great interest in the condensed matter physics community.1,2,3,4,5 One key to making progress in this area is the identification of model material systems that are complex enough to exhibit the physics of interest but simple enough to be compared to theory. Studies of several low dimensional magnetic materials have greatly improved our understanding of QCP physics in insulating solids,6,7,8 but our understanding of QCP phenomena in metallic systems is much less clear.2, 9, 10 One hindrance is that few systems are microscopically homogeneous near the QCP, due to the need for small concentrations of dopants, and this can disrupt the feedback loop between theory and experiment.
The NiCoCrx alloys (with x ≈ 1) are related to the recently discovered high-entropy-alloys, such as NiCoCrFeMn,11, 12 where configurational entropy stabilizes a random distribution of elements on a simple face-centered cubic lattice (fcc). These alloys have been shown to be chemically homogeneous from the centimeter to the nanometer scale. An example of an atomic resolution scanning transmission electron microscope (STEM) image from a NiCoCr single crystal is shown in Fig. 1. No clustering of any of the three elements is observed. Energy dispersive spectroscopy data for the NiCoCr0.8 and NiCoCr crystals presented in Fig. S1 illustrate chemical homogeneity on the micron scale.
The high-entropy-alloys have remarkable mechanical properties,13 and enhanced resistance to radiation damage.14 The alloys are also unusual because all elements have roughly equal concentrations and the dilute impurity approximation is invalid. A recent theoretical model based on enthalpies of formation coupled with entropy stabilization successfully predicts which combination of elements will form a single-phase solid solution.15 The composition NiCoCr also has remarkable mechanical properties at cryogenic temperatures and has been termed a medium-entropy alloy.16
Here we report low temperature magnetization and heat capacity for single crystals of the concentrated solid solutions NiCoCrx with x ≈ 0.8 as a function of temperature and magnetic field. The extreme chemical disorder in these alloys is an essential component of the physics, suppressing a first order ferromagnetic transition that invariably occurs in many other potential ferromagnetic quantum critical point (FQCP) systems.9 In carefully prepared single crystals with electropolished surfaces and minimum mechanical damage, we find that the BKV theory17, 18 of metallic ferromagnetic QCPs provides an excellent description of all of our data. The approximate temperature–magnetic field phase diagram that describes the boundary between the quantum critical and the Fermi liquid regions is also determined for the NiCoCr0.8 composition from magnetization data. Finally we demonstrate that the magnetic Gruneisen parameter [Γ = (dM/dT)/Cp] diverges, another indication of a FQCP. These findings open the door to the study of QCP physics in a new and flexible class of model materials.
Results
As is illustrated in Fig. 2, NiCoCr0.8 is close to the Cr concentration where the ferromagnetic transition temperature, T c, goes to zero. The ferromagnetism is highly itinerant for all compositions x > 0.5; for values of x ≈ 0.6 the saturation moment per atom is 0.15 μB, T c = 75 ± 2 K, and the Rhodes–Wohlfarth ratio is 8.5.19 For increasing Cr concentrations, x, there is a rapid nearly exponential decrease in T c, and in addition to the rapid depression of T c, increasing the Cr concentration introduces a type of frustration since the spins on the Cr atoms want to be antiparallel to neighboring Ni, Co, and Cr spins, which cannot be satisfied on a fcc lattice. For Cr concentrations greater than x ≈ 0.4, theory (KKR–CPA)14, 19, 20 greatly over-estimates both the spontaneous magnetic moments and T c (see Fig. 3c. Ref. 19) indicating significant effects that are not captured by density functional theory. This deviation from theory and the observation of T c → 0 K motivate a detailed characterization of the critical behavior and exponents for compositions near NiCoCr0.8.
The low field and low temperature magnetization data from a NiCoCr0.8 single crystal are shown in Fig 3a. The divergence of the low field susceptibility, χ = (M/H) as T approaches zero determines the critical exponent γ T , where χ α T −γT. For low applied magnetic fields from 0.0001 to 0.1 Tesla, the susceptibility of NiCoCr0.8 crystals diverge with γ T ≈ 0.5. It is important to note that to observe this behavior the surface of the crystals had to be electropolished and carefully cleaned before magnetic measurements (see experimental section), otherwise the magnetic response is dominated by extrinsic defects.
The critical exponent, δ, is determined from the critical isotherm in the limit that T→0 K and is defined as M(T,H) α H1/δ. Although technically δ is only defined at T c (which is 0 at the FQCP), the values are expected to saturate at a low enough temperature. Magnetization isotherms at our lowest temperature of 1.8 K are displayed in Fig. 3b for two NiCoCr0.8 crystals. Power law fits to these data yield an estimate of δ = 1.55 ± 0.1.
The critical exponent, β T, can be formally associated with a FQCP and is defined in terms of the order parameter M in the limit of H→0 and T→0 as proportional to \(T^{\beta_{\rm T}}\). The proportionality constant is of course zero at the FQCP since the spontaneous magnetization is only non-zero in the ordered magnetic phase. In this case the value of βT can be estimated by tuning the Cr composition slightly away from the critical value of x ≈ 0.8 and into the ferromagnetic state (x < 0.8, see Fig. 2) and then determine the values of βT as x = 0.8 is approached. This was done for 3 polycrystalline NiCoCrx samples with x = 0.75, 0.70, 0.65 and Tc’s of 4.4, 16.5, and 33.5 K. These values for T c were defined by the maximum in -dM/dT with H = 100 Oe. This definition of T c appeared to be the least subject to assumptions. We note however that changing T c by a couple of degrees did not significantly effect the results for β T. The evolution of the magnetization below T c is evaluated for each of the three alloys with a small applied field of H = 100 Oe. For each alloy the magnetization data below T c was fit to a power law for various values of T c-T, with the assumption that the correct value of β T is attained as T c-T approaches zero (Fig. 3c). Using this analysis, we obtain the value of β T for each alloy. These values are plotted in Fig. 3d and yield an approximate value for β T (x ≈ 0.8) of 1.
Taken together, direct analysis of the magnetization data for NiCoCr0.8 give the following critical exponents: γ T ≈ 0.5, δ ≈ 3/2, and β T ≈ 1. The subscript T for γ and β indicate the temperature dependence of the relevant variables when the composition x has been tuned to its critical value (x cr ≈ 0.8) where the transition temperature is zero. This is also the notation used in the BKV theory17, 18. These experimental values for γ T , δ, and β T are also the values predicted from BKV theory in the “dirty” or high-disordered limit. In this limit the Widom relationship γ T = β T (δ − 1), is expected to hold18 and within our experimental error it does. This means that only two of the three exponents are independent. In the “dirty” limit, BKV theory18 also predicts that the Curie temperature should increase rapidly as the composition is tuned away from the critical composition as \({T_c} \propto {\left( {{x_c} - x} \right)^{\nu{z_m}}} = {\left( {{x_c} - x} \right)^2}\), since zm = 2 and ν = 1 (from the theory). Although we do not have enough experimental data to show that z m ν is exactly two, a force fit of the T c data to (x c − x)2 does provide a good description of our data as shown in Fig. 3d.
One test of quantum critical behavior is the divergence of the magnetic Gruneisen parameter21 (also called the magnetocaloric ratio), Γ = (dM/dT)/Cp, as T approaches 0.10, 22, 23 Low temperature heat capacity data from NiCoCr0.8 with H = 0 are displayed in Fig. 4a. At the lowest temperatures C p is proportional to T α with a value of α = 0.92, which implies that C p/T weakly diverges as T −0.08. This weak divergence of C p/T is close to the exponent value of \({\bar {\rm \alpha}}\) T = 0 expected from BKV theory. As noted above, the low field (H = 100 Oe) susceptibility (M/H), diverges as ≈ T −0.5, which implies that dM/dT diverges as ≈ T −3/2. Thus Γ diverges as ≈ T −2.5. This low temperature divergence is demonstrated in Fig. 4b, using either the total measured heat capacity or just the electronic/magnetic portion. For the three ferromagnetic compositions (x = 0.75, 0.7, 0.65), the low temperature magnetic Gruneisen parameter approaches 0 since in low applied magnetic fields dM/dT→0.
The scaling of the magnetization data, M(T, H), are determined by the critical exponents.22, 24 All of the magnetization data from the NiCoCr0.8 single crystals should collapse onto a single curve when \({\rm M/T}^{\beta_{\rm T}}\) is plotted vs. H/\({\rm T}^{\,\gamma_{\rm T}+\beta_{\rm T}}\). Since γ T and β T for NiCoCr0.8 are found experimentally to have the values of γ T ≈ 0.5, and β T ≈ 1, this implies that if M(T, H)/T is plotted vs. H/T1.5, all of the magnetization data should fall on one curve. This is indeed the case as illustrated in Fig. 5. It is important to emphasize that we are not using the scaling plots to extract precise values for γ T and β T, but rather we simply illustrate that the values found do result in excellent scaling. It has been demonstrated recently that it is very difficult to extract unique values of critical exponents from scaling plots alone.22
Discussion
We have provided strong evidence that NiCoCr0.8 is close to a FQCP and the experimentally determine critical exponents (γ T ≈ 0.53 ± 0.02, β T ≈ 1 ± 0.02, δ ≈ 1.55 ± 0.1, νz m = 2, \({\bar {\rm \alpha}}\) T = 0.08) are close to the values expected from BKV theory in the asymptotic limit of high-disorder (γ T = 1/2, β T = 1, δ = 3/2, νz m = 2, \({\bar {\rm \alpha}}\) T = 0.0). In the limit of high disorder, effects due to Griffiths phases are possible particularly in the paramagnetic region. The BKV theory18 does not consider Griffith phases and the good agreement between our data and the BKV model would suggest that Griffith phase effects are small in the NiCoCr0.8 alloys. This is also the conclusion from ac susceptibility measurements shown in Figure S2.
Magnetic field, which can be thought of as another variable perpendicular to the temperature and Cr composition axis (Fig. 2), is used to probe the extent of the quantum critical region away from the FQCP of the NiCoCr0.8 alloy. As a magnetic field is applied to NiCoCr0.8 the magnetic fluctuations are suppressed and the system is tuned toward a normal Fermi liquid with a Pauli susceptibility. This behavior suggests there is a region where the system’s behavior is governed by quantum fluctuations, as opposed to thermally-induced fluctuations, consistent with the existence of quantum critical behavior in NiCoCr0.8. This crossover from the quantum critical regime to the Fermi liquid regime occurs at T* for a given applied magnetic field B. Experimentally the crossover temperature T* is usually estimated as the temperature where a maximum occurs in χ(T), with χ = dM/dH extracted from DC magnetization curves at fixed temperatures or from ac susceptibility data10, 23 such as shown in Fig. 6a. The dependence of T* on applied magnetic field B is shown in Fig. 6b. A power law fit to T* vs B gives T* = 5.62 B0.67 (or T*1.5 vs B) the same scaling variable used in Fig. 5. This provides another self-consistency check that the analysis of the magnetization data from NiCoCr0.8 in terms of a FQCP is reasonable.
As noted previously in reference,19 the residual resistivity, ρ 0, of the NiCoCrx alloys is high, of the order of 80 μΩ cm, and for many of the alloys with x ≈ 0.8 the resistivity is linear down to at least 1.0 K. A large value of ρ 0 is not considered in the BKV analysis presented in reference,18 and for that reason resistivity data were not discussed in the main portion of the manuscript. However, the electrical resistivity of a NiCoCr0.8 crystal in various magnetic fields is shown in Figure S3, along with the temperature-magnetic field scaling behavior of the data, which is close to that presented in Fig 6.
As briefly discussed in the experimental section, strain can also be used as another tuning variable, though we have not developed a strain-based phase diagram to complement Fig. 6b.
Since there are well over two hundred ternary concentrated solid solutions with compositions ABC predicted to form,15 this class of entropy stabilized alloys represents an interesting new class of tunable materials for the investigation of quantum critical behavior.
Methods
Single crystals of NiCoCr0.8 are grown in an optical floating zone furnace. Polycrystalline rods for the crystal growth are produced by arc-melting stoichiometric amounts (1:1:0.8) of high purity (>99.9%) Ni, Co, and Cr in an argon atmosphere followed by drop casting into a copper mold.25 Laue back reflection and X-ray diffraction measurements indicated single crystals with a small mosaic spread (≈0.4°). Crystals suitable in size for magnetization (≈200 mg) or heat capacity (≈25 mg) are cut from a large single crystal boule using an electro-discharge machine. To eliminate the damage layer created after cutting or sanding, the crystals are electropolished in an 85% H3PO4 solution with approximately a 10 V bias. This surface preparation is essential, particularly for the magnetic measurements. If, for example, the samples are cut with a low speed diamond saw and then polished with SiC sandpaper, or just cut with a diamond saw, it is found that the temperature dependence and magnitude of the low field magnetic susceptibility is dramatically altered- by more than a factor of two. We also find that severe coldworking the material increases the low temperature susceptibility by about a factor of ten. These results suggest that strain and pressure may also be a useful QCP tuning parameter for the NiCoCr0.8 alloys, although that is beyond the scope of the present work. The chemical homogeneity of crystals grown this way have been checked on the micron scale using energy dispersive spectroscopy measurements and on the nanoscale using atom probe measurements.25 To further examine both the crystalline perfection and the chemical homogeneity we perform atomic resolution STEM measurements and single column electron energy loss spectroscopy (EELS) on a NiCoCr single crystal. The STEM results, shown in Fig. 1, confirm the crystalline perfection and the absence of clustering of any of the elements.
The DC and AC magnetic measurements are made using a commercial SQUID magnetometer (Magnetic Property Measurement System from Quantum design) and the ac susceptibility option of a Physical Property Measurement System (PPMS) (also from Quantum Design), respectively. To check for possible glassy magnetic behavior, both zero-field-cooled and field-cooled measurements are performed. Within experimental error there is no difference in the magnetization data at all fields. As a further check, ac susceptibility measurements are made as a function of frequency, and no glassy behavior could be detected (Figure S2). Heat capacity measurements are made using the PPMS heat capacity option. Identification of commercial equipment does not imply endorsement by National Institute of Standards and Technology.
Data availability
All relevant data are available from the authors.
References
Sachdev, S. Quantum Phase Transitions. (Cambridge University Press: NY, 2011).
Sachdev, S. & Keimer, B. Quantum criticality, Phys. Today 64, 29–35 (2011).
Analytis, J. G., Kuo, H.-H., McDonald, R. D., Wartenbe, M., Rourke, P. M. C., Hussey, N. E. & Fisher, I. R. Transport near a quantum critical point in BaFe2(As1-xPx) 2 . Nat. Phys 10, 194–197 (2014).
Moriya, T. & Ueda, K. Spin fluctuations and high temperature superconductivity. Adv. Phys 49, 555–606 (2000).
Marel, Dvd et al. Quantum critical behavior in a high-temperature superconductor. Nature. 425, 271–274 (2003).
Coldea, R. et al. Quantum criticality in an ising chain: experimental evidence for emergent E-8 symmetry. Science 327, 177–180 (2010).
Kinross, W., Fu, M., Munsie, T. J., Dabkowska, H. A., Luke, G. M., Sachdev, S. & Imai, T. Evolution of quantum fluctuations near the quantum critical point of the transverse field ising chain system Co2Nb2 O 6 . Phys. Rev. X 4, 031008 (2014).
Ruegg, C. et al. Quantum magnets under pressure: controlling elementary excitations in TlCuCl3. Phys. Rev. Lett. 100, 205701 (2008).
Brando, M., Belitz, D., Grosche, F. M. & Kirkpatrick, T. R. Metallic quantum ferromagnets. Rev. Mod. Phys. 88, 025006 (2016).
Matsumoto, Y. et al. Quantum criticality without tuning in the mixed valence compound β-YbAlB4. Science 331, 316–319 (2011).
Cantor, B., Chang, I. T. H., Knight, P. & Vincent, A. J. B. Microstructural development in equiatomic multicomponent alloys. Mat. Sci. Eng. A 375-377, 213–218 (2004).
Wu, Z., Bei, H., Otto, F., Pharr, G. M. & George, E. P. Recovery, recrystallization, grain growth and phase stability of a family of FCC-structured multicomponent equiatomic solid solution alloys. Intermetallics 46, 131–140 (2014).
Gludovatz, B., Hohenwarter, A., Catoor, D., Chang, E. H., George, E. P. & Ritchie, R. O. A fracture-resistant high-entropy alloy for cryogenic applications. Science 345, 1153–1158 (2014).
Zhang, Y. et al. Influence of chemical disorder on energy dissipation and defect evolution in nickel and nickel-based concentrated solid solution alloys. Nat. Comm 6, 8736–8744 (2015).
Troparevsky, C. M., Morris, J. R., Kent, P. R. C., Lupini, A. R. & Stocks, G. M. Criteria for predicting the formation of single-phase high-entropy alloys. Phys. Rev. X 5, 011041 (2015).
Gludovatz, B. et al. Exceptional damage-tolerance of a medium-entropy alloy CrCoNi at cryogenic temperatures. Nat. Comm. 7, 10602 (2016).
Belitz, D., Kirkpatrick, T. R. & Vojta, T. First order transitions and multicritical points in weak itinerant ferromagnets. Phys. Rev. Lett. 82, 4707–4710 (1999).
Kirkpatrick, T. R. & Belitz, D. Exponent relations at quantum phase transitions with applications to metallic quantum ferromagnets. Phys. Rev. B 91, 214407 (2015).
Sales, B. C., Jin, K., Bei, H. B., Stocks, G. M., Samolyuk, G. D., May, A. F. & McGuire, M. A. Quantum critical behavior in a concentrated ternary solid solution. Sci. Rep. 6, 26179 (2016).
Stocks, G. M., Temmerman, W. N. & Gyorffy, B. L. Complete solution of the korringa-kohn-rostoker coherent-potential-approximation equations: cu-ni alloys. Phys. Rev. Lett. 41, 339–343 (1978).
Zhu, L., Garst, M. & Rosch, A. Q. Si Universally diverging grüneisen parameter and the magnetocaloric effect close to quantum critical points. Phys. Rev. Lett. 91, 066404 (2003).
Huang, K. et al. Quantum critical scaling in the disordered itinerant ferromagnet UCo1-xFexGe. Phys. Rev. Lett. 117, 237202 (2016).
Wu, L. S., Kim, M. S., Park, K., Tsvelik, A. M. & Aronson, M. C. Quantum critical fluctuations in layered YFe2Al10. Proc. Natl Acad. Sci. 111, 14088 (2014).
Huang, C. L. et al. Anomalous quantum criticality in an itinerant ferromagnet. Nat. Commun. 6, 8188 (2015).
Wu, Z., Gao, Z., Bei, Y. F. & Single-crystal, H. Plastic behavior of a single-phase, face-centered-cubic structured, equiatomic FeNiCrCo. Scr. Mater. 109, 108–112 (2015).
Acknowledgements
It is a pleasure to acknowledge helpful discussions with Jamie Morris, Lekh Poudel, Malcolm Stocks, German Samolyuk, Claudia Troparevsky and Jiaqiang Yan. I particularly want to thank Claudia Troparevsky for providing an extensive list of predicted ternary solid solution alloys using the methodology developed in reference15. This research was supported primarily by the Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division (B.C.S., J.N., A.F.M., M.F.C., M.A.M.). K.J. and H.B. were supported by the Energy Dissipation to Defect Evolution (EDDE), an Energy Frontier Research Center funded by the U. S. Department of Energy, Office of Science, BES. Research at ORNL’s Spallation Neutron Source was supported by the Scientific User Facilities Division, Office of Basic Energy Sciences, U S. Department of Energy (A.C.).
Author information
Authors and Affiliations
Contributions
B.C.S. conceived the experiments, and made most of the transport, magnetic, and heat capacity measurements with help from A.F.M., M.A.M., and J. N. K. J. and H. B. grew, oriented, cut, and electropolished the crystals. M.F.C. took and analyzed atomic resolution STEM and EELS data. A.C. and N.B. made key contributions to the analysis of the data. All authors contributed to the preparation of the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing financial interests.
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Sales, B.C., Jin, K., Bei, H. et al. Quantum critical behavior in the asymptotic limit of high disorder in the medium entropy alloy NiCoCr0.8 . npj Quant Mater 2, 33 (2017). https://doi.org/10.1038/s41535-017-0042-7
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41535-017-0042-7
- Springer Nature Limited
This article is cited by
-
Data-driven electron-diffraction approach reveals local short-range ordering in CrCoNi with ordering effects
Nature Communications (2022)
-
Quantum-critical scale invariance in a transition metal alloy
Communications Physics (2020)